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thedarktiger

Member

Registered: 2014-01-10

Posts: 91

dilation of a triangle (whatever that means)

Equilateral triangle ABC  has centroid G. Triangle A'B'C' is the image of triangle ABC upon a dilation with center G and scale factor -2/3. Let K be the area of the region that is within both triangles. Find K/[ABC].

Meh. To many fancy words.

thanks!

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Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: dilation of a triangle (whatever that means)

hi

A 'dilation' is what I would call an 'enlargement'.  It means every length of the shape is changed by a fixed scale factor.  It is still called a dilation(enlargement) even if the scale factor is a fraction, as here.  So the new shape is another equilateral triangle, 2/3 of the size.  If you join each point of the original to the corresponding point of the new image, these lines cross at a point called the centre of enlargement.  In this example, this point is at G, one third of the way up the medians.  A minus scale factor means the corresponding points are on the other side of the centre.  This has the effect of reversing the 'sense' of the shape, by which I mean if A to B to C is a clockwise movement, the A' to B' to C' will be an anticlockwise movement.

On MIF it is called resizing:  http://www.mathsisfun.com/geometry/resizing.html

The centroid is one third of the way up from the midpoint of a side, and the length of the corresponding median in the new shape is 2/3 of the original.  That should help to fix the dimensions of the new shape compared with the original and also the size of those little equilateral triangles that are outside the overlap. 

One more fact and you should be able to finish this problem:  If a shape is enlarged by a length scale factor 'f' then the area will change by a factor of f squared.  So the new triangle has an area 4/9 of the original for example.

Bob

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